How to render certain edges
You may use the Edge Mark option of the Freestyle to define what edges would be rendered.


Enable Freestyle renderer in Render header-->Freestyle panel. Set the line thickness as you like.


Select the edges you want the Freestyle to render, then press Ctrl+E-->Mark Freestyle Edge to make them renderable.


In Render Layers header-->Freestyle Line Set panel check the Edge Mark checkbox. Uncheck the Silhouette, Border and Crease checkboxes to make the contour of the object renderless. 


Here's the render result:


You can go further and render only selected edges. With the vertices/edges selected in Edit Mode duplicate them with Shift+D and make them a separate object (P-->Selection). Move the original object to another layer and hit render. 


Here's the result:


Not sure if it is what you're looking for. Let me know if I helped you.

UPDATE: If you want to render only the edges give the object a Transparent shader in Cycles. The Freestyle settings should stay the same as in the examples above.